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Parikh vectors
8PJS_1 4NOB_1 4DIQ_1 Letter Amino acid
18 8 35 E Glutamic acid
43 12 30 G Glycine
12 7 10 I Isoleucine
12 9 64 L Leucine
6 6 29 D Aspartic acid
13 1 13 H Histidine
0 3 7 K Lycine
7 1 11 M Methionine
31 16 25 S Serine
12 9 23 T Threonine
42 7 31 V Valine
24 3 43 A Alanine
6 5 41 R Arginine
0 7 26 Q Glutamine
6 1 8 W Tryptophan
12 9 14 N Asparagine
0 4 5 C Cysteine
0 5 21 F Phenylalanine
24 5 33 P Proline
12 8 20 Y Tyrosine 

8PJS_1|Chain A|SAKeCEref|synthetic construct (32630)
>4NOB_1|Chain A|Polymeric immunoglobulin receptor|Mus musculus (10090)
>4DIQ_1|Chains A, B|Lysine-specific demethylase NO66|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
8PJS , Knot 31 280 0.20 32 44 50 
GSHMGGLIYAVGGDGHTHLSSVEAYNPETNEWSPVAPMSVPRSGVGVAVIGGLIYAVGGDGHTHLSSVEAYNPETNEWSPVAPMSVPRSGVGVAVIGGLIYAVGGDGHTHLSSVEAYNPETNEWSPVAPMSVPRSGVGVAVIGGLIYAVGGDGHTHLSSVEAYNPETNEWSPVAPMSVPRSGVGVAVIGGLIYAVGGDGHTHLSSVEAYNPETNEWSPVAPMSVPRSGVGVAVIGGLIYAVGGDGHTHLSSVEAYNPETNEWSPVAPMSVPRSGVGVAVI
4NOB , Knot 65 126 0.83 40 103 123 
QDYGGSPIFGPQEVSSIEGDSVSITCYYPDTSVNRHTRKYWCRQGASGMCTTLISSNGYLSKEYSGRANLINFPENNTFVINIEQLTQDDTGSYKCGLGTSNRGLSFDVSLEVSQVPELAENLYFQ
4DIQ , Knot 199 489 0.84 40 232 456 
LRRRYTMASGPQVDNTGGEPAWDSPLRRVLAELNRIPSSRRRAARLFEWLIAPMPPDHFYRRLWEREAVLVRRQDHTYYQGLFSTADLDSMLRNEEVQFGQHLDAARYINGRRETLNPPGRALPAAAWSLYQAGCSLRLLCPQAFSTTVWQFLAVLQEQFGSMAGSNVYLTPPNSQGFAPHYDDIEAFVLQLEGRKLWRVYRPRAPTEELALTSSPNFSQDDLGEPVLQTVLEPGDLLYFPRGFIHQAECQDGVHSLHLTLSTYQRNTWGDFLEAILPLAVQAAMEENVEFRRGLPRDFMDYMGAQHSDSKDPRRTAFMEKVRVLVARLGHFAPVDAVADQRAKDFIHDSLPPVLTDRERALSVYGLPIRWEAGEPVNVGAQLTTETEVHMLQDGIARLVGEGGHLFLYYTVENSRVYHLEEPKCLEIYPQQADAMELLLGSYPEFVRVGDLPCDSVEDQLSLATTLYDKGLLLTKMPLALNAENLYFQ

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(8PJS_1)}(2) \setminus P_{f(4NOB_1)}(2)|=30\), \(|P_{f(4NOB_1)}(2) \setminus P_{f(8PJS_1)}(2)|=89\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1001111101111010001001010010000101111101100111111111110111101000100101001000010111110110011111111111011110100010010100100001011111011001111111111101111010001001010010000101111101100111111111110111101000100101001000010111110110011111111111011110100010010100100001011111011001111111
Pair \(Z_2\) Length of longest common subsequence
8PJS_1,4NOB_1 119 2
8PJS_1,4DIQ_1 210 3
4NOB_1,4DIQ_1 197 7

Newick tree

 
[
	4DIQ_1:11.41,
	[
		8PJS_1:59.5,4NOB_1:59.5
	]:52.91
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{406 }{\log_{20} 406}-\frac{126}{\log_{20}126})=84.6\)
Status Protein1 Protein2 d d1/2
Query variables 8PJS_1 4NOB_1 57 39.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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